Place 3/4 cup unsalted butter in a light-colored skillet over medium heat. Cook, stirring, as it melts, foams, and turns amber with brown specks, 5–8 minutes. Immediately scrape into a shallow, heatproof dish and chill until opaque but pliable, 25–35 minutes.

Preheat oven to 325°F. Grease two 9-inch round cake pans, line bottoms with parchment, and dust sides with cocoa or flour; tap out excess.

Grate zucchini on the large holes of a box grater. Wrap in a clean towel and twist firmly to expel as much liquid as you can. Measure 2 1/4 cups firmly packed; fluff with a fork and set aside.

In a large bowl, whisk together flour, cocoa, baking powder, baking soda, salt, cinnamon, and espresso powder.

In the bowl of a stand mixer (or with a hand mixer), beat the eggs, egg yolk, and sugar on medium-high until pale and slightly thickened, 2–3 minutes. With the mixer on low, slowly stream in the oil, then blend in vanilla and sour cream until smooth.

Whisk the hot coffee with 2 tablespoons of the cocoa mixture to form a smooth slurry. Mix this into the wet ingredients to bloom the cocoa flavors.

Add the remaining dry ingredients in two additions on low speed, mixing just until a few flour streaks remain. Fold in the zucchini by hand until evenly distributed. Let the batter rest 5 minutes to hydrate the cocoa.

Divide batter evenly between pans and smooth tops. Bake 32–36 minutes, until a tester comes out with a few moist crumbs. Cool 15 minutes in pans, then turn out to a rack to cool completely.

Beat the softened 1/2 cup butter in a mixing bowl until creamy, 1 minute. Add the chilled brown butter and whip on medium-high until pale and fluffy, 2–3 minutes.

Sift cocoa with 2 cups powdered sugar. Mix into the butter on low until combined. Add vanilla and salt. Beat in remaining powdered sugar alternately with milk, 1 tablespoon at a time, until smooth and spreadable. Whip 2 more minutes for extra lightness.

Level cooled cakes if needed. Place one layer on a board and spread about 1 heaping cup frosting over the top. Set the second layer on, cut side down. Apply a thin crumb coat and chill 10–15 minutes. Finish with a generous final coat, using an offset spatula to create loose swoops. Slice and serve.